Problem: Determine how many solutions exist for the system of equations. ${8x+2y = 18}$ ${12x+3y = 27}$
Explanation: Convert both equations to slope-intercept form: ${8x+2y = 18}$ $8x{-8x} + 2y = 18{-8x}$ $2y = 18-8x$ $y = 9-4x$ ${y = -4x+9}$ ${12x+3y = 27}$ $12x{-12x} + 3y = 27{-12x}$ $3y = 27-12x$ $y = 9-4x$ ${y = -4x+9}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -4x+9}$ ${y = -4x+9}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${8x+2y = 18}$ is also a solution of ${12x+3y = 27}$, there are infinitely many solutions.